Quantum Mechanics of Charged Particle Beam Optics Multidisciplinary and Applied Optics SeriesEditors: VasudevanLakshminarayanan,UniversityofWaterloo,Ontario,Canada HassenGhalila,UniversityTunisElManar,Tunisia AhmedAmmar,UniversityTunisElManar,Tunisia L.SrinivasaVaradharajan,UniversityofWaterloo,Ontario,Canada UnderstandingOpticswithPython VasudevanLakshminarayanan,HassenGhalila,AhmedAmmar,L.Srinivasa Varadharajan Formoreinformationaboutthisseries,pleasevisit: https://www.crcpress.com/Multidisciplinary-and-Applied-Optics/book-series/ CRCMULAPPOPT Quantum Mechanics of Charged Particle Beam Optics Understanding Devices from Electron Microscopes to Particle Accelerators RamaswamyJagannathanandSameenAhmedKhan CRCPress Taylor&FrancisGroup 6000BrokenSoundParkwayNW,Suite300 BocaRaton,FL33487-2742 ©2019byTaylor&FrancisGroup,LLC CRCPressisanimprintofTaylor&FrancisGroup,anInformabusiness NoclaimtooriginalU.S.Governmentworks Printedonacid-freepaper InternationalStandardBookNumber-13:978-1-138-03592-8(Hardback) Thisbookcontainsinformationobtainedfromauthenticandhighlyregardedsources. 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VisittheTaylor&FrancisWebsiteat http://www.taylorandfrancis.com andtheCRCPressWebsiteat http://www.crcpress.com Dedication Tothememoryofmyparents Thirali Aravamudha Ramaswamy Iyengar & Andal Ramaswamy —–RamaswamyJagannathan Tothememoryofmyparents Hamid Ahmed Khan & Nighat Hamid Khan —–SameenAhmedKhan Contents Preface.......................................................................................................................xi Authors.....................................................................................................................xv Chapter1 Introduction.....................................................................................1 Chapter2 AnIntroductoryReviewofClassicalMechanics............................9 2.1 SingleParticleDynamics........................................................9 2.1.1 LagrangianFormalism...............................................9 2.1.1.1 BasicTheory..............................................9 2.1.1.2 Example:MotionofaCharged ParticleinanElectromagneticField........10 2.1.2 HamiltonianFormalism...........................................12 2.1.2.1 BasicTheory............................................12 2.1.2.2 Example:MotionofaCharged ParticleinanElectromagneticField........14 2.1.3 HamiltonianFormalisminTermsofthePoisson Brackets....................................................................16 2.1.3.1 BasicTheory............................................16 2.1.3.2 Example:DynamicsofaCharged ParticleinaConstantMagneticField.....18 2.1.4 ChangingtheIndependentVariable.........................21 2.1.4.1 BasicTheory............................................21 2.1.4.2 Example: Dynamics of a Charged ParticleinaConstantMagneticField.....22 2.1.5 CanonicalTransformations......................................25 2.1.5.1 BasicTheory............................................25 2.1.5.2 Optical Hamiltonian of a Charged Particle Moving Through an ElectromagneticOpticalElementwith aStraightAxis.........................................29 2.1.6 SymplecticityofCanonicalTransformations..........31 2.1.6.1 Time-Independent Canonical Transformations.......................................31 2.1.6.2 Time-DependentCanonicalTransfor- mations:HamiltonianEvolution..............32 2.1.6.3 CanonicalInvariants:PoissonBrackets...35 2.2 DynamicsofaSystemofParticles.......................................36 vii viii Contents Chapter3 AnIntroductoryReviewofQuantumMechanics..........................41 3.1 Introduction...........................................................................41 3.2 GeneralFormalismofQuantumMechanics.........................42 3.2.1 Single Particle Quantum Mechanics: FoundationalPrinciples...........................................42 3.2.1.1 QuantumKinematics...............................42 3.2.1.2 QuantumDynamics.................................51 3.2.1.3 DifferentPicturesofQuantum Dynamics.................................................64 3.2.1.4 Ehrenfest’sTheorem................................68 3.2.1.5 Spin..........................................................70 3.3 NonrelativisticQuantumMechanics....................................78 3.3.1 NonrelativisticSingleParticleQuantum Mechanics................................................................78 3.3.1.1 FreeParticle.............................................78 3.3.1.2 LinearHarmonicOscillator.....................90 3.3.1.3 Two-DimensionalIsotropicHarmonic Oscillator...............................................100 3.3.1.4 ChargedParticleinaConstant MagneticField.......................................103 3.3.1.5 ScatteringStates....................................107 3.3.1.6 ApproximationMethods, Time-DependentSystems,andthe InteractionPicture.................................111 3.3.1.7 Schro¨dinger–Pauli Equation for the Electron..................................................117 3.3.2 Quantum Mechanics of a System of Identical Particles..................................................................119 3.3.3 PureandMixedStates:DensityOperator..............126 3.4 RelativisticQuantumMechanics........................................132 3.4.1 Klein–GordonEquation.........................................132 3.4.1.1 Free-ParticleEquationand DifficultiesinInterpretation..................132 3.4.1.2 Feshbach–VillarsRepresentation..........137 3.4.1.3 Charged Klein–Gordon Particle in a ConstantMagneticField........................139 3.4.2 DiracEquation.......................................................141 3.4.2.1 Free-ParticleEquation...........................141 3.4.2.2 Zitterbewegung......................................149 3.4.2.3 SpinandHelicityoftheDirac Particle...................................................150 3.4.2.4 SpinMagneticMomentoftheElectron andtheDirac–PauliEquation................153 3.4.2.5 ElectroninaConstantMagneticField...154 Contents ix 3.4.3 Foldy–WouthuysenTransformation.......................156 3.4.3.1 Foldy–WouthuysenRepresentationof theDiracEquation.................................156 3.4.3.2 Foldy–WouthuysenRepresentationof the Feshbach–Villars form of the Klein–GordonEquation.........................168 3.5 Appendix: The Magnus Formula for the Exponential SolutionofaLinearDifferentialEquation.........................170 Chapter4 AnIntroductiontoClassicalChargedParticleBeamOptics......173 4.1 Introduction: Relativistic Classical Charged Particle BeamOptics........................................................................173 4.2 FreePropagation.................................................................174 4.3 OpticalElementswithStraightOpticAxis........................178 4.3.1 Axially Symmetric Magnetic Lens: Imaging in ElectronMicroscopy..............................................178 4.3.2 NormalMagneticQuadrupole...............................197 4.3.3 SkewMagneticQuadrupole...................................202 4.3.4 AxiallySymmetricElectrostaticLens...................204 4.3.5 ElectrostaticQuadrupole........................................205 4.4 Bending Magnet: An Optical Element with aCurvedOpticAxis...........................................................206 4.5 NonrelativisticClassicalChargedParticle BeamOptics........................................................................210 Chapter5 QuantumChargedParticleBeamOptics:ScalarTheoryfor Spin-0andSpinlessParticles.......................................................213 5.1 GeneralFormalismofQuantumChargedParticle BeamOptics........................................................................213 5.2 Relativistic Quantum Charged Particle Beam Optics BasedontheKlein–GordonEquation................................214 5.2.1 GeneralFormalism.................................................214 5.2.2 FreePropagation:Diffraction................................230 5.2.3 Axially Symmetric Magnetic Lens: Electron OpticalImaging.....................................................232 5.2.3.1 ParaxialApproximation:Point-to-Point Imaging..................................................232 5.2.3.2 Going Beyond the Paraxial Approximation:Aberrations..................250 5.2.3.3 QuantumCorrectionstotheClassical Results...................................................260 5.2.4 NormalMagneticQuadrupole...............................262 5.2.5 SkewMagneticQuadrupole...................................267 5.2.6 AxiallySymmetricElectrostaticLens...................269